Problem

Graph the solutions of each system. See Example 4…

06:57

Problem 28 Medium Difficulty

Graph the solutions of each system. See Example 4.
$$
\left\{\begin{array}{l}
{x-y \leq 6} \\
{x+2 y \leq 6} \\
{x \geq 0}
\end{array}\right.
$$

Answer

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Algebra

Elementary and Intermediate Algebra

Chapter 4

Systems of Linear Equations and Inequalities

Section 5

Solving Systems of Linear Inequalities

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Video Transcript

we have the system off inequalities X minus Y let's or anybody at six x plus two y less than or equal six and X very or wouldn't see. So we started wearing the question of the boundary lines for the first inequality. The boundary line has equation X minus y equal to six, and it's a solid line for the second inequality. The boundary line has equation X Plus two. Why do you want to six and it's solid on this. Use another quarter for this one. And for the third inequality, that bounder Reliant House equation X zero. And again, this is so now this line is just the vertical line that go through the value X equal to see room. So that means that this is the Y axis. Okay, Now, in order to graph the other two boundary lying's, we're gonna look at the X and why Intercepts for the 1st 1 x minus way what you seem. Six. The X intercept is the point with why coordinated will to cereal. So if we're replacing the line equation, we will get in this case at X is equal to six. So the X intercept disappoint 60 Now the Y intercept Issa Point, whose X coordinate is equal to zero. And if we replace in the equation, we get minus y he went to six. That is the same that say, why equal to minus six? So the why intercept for the first Bonder Relying is the point, Cyril minus six Now for the second round of relying Hi, that has equation X plus two. Why? Able to six? Okay, the X intercept is the point with White coordinated equal to zero. And if we replace in the equation, we will get X equal to six. So the X intercept Issa 0.60 and the Y intercept is the point whose X coordinate this equal to zero. And if we replace in the question with toe white 1 to 6 so why is able to three? So the Y intercept Issa 0.3 Now, with this information, we can draw the boundary Reliance sublet goals with the graph. So we have the why and the x axis we will draw first. The third bone. They're relying that were really said that it's a vertical line that go through the body acceptable to see row So that means that is the Y axis. Now we can go with the second bomb, the relying. And in order to do that, we start by graph in the to X and y intercept. So first is the point. Uh 60 Let's say that is this point here and the 0.3 So must be a point like this. So the boundary line should be this life now, uh, second Boulder relying we have the X and Y intercept the 0.6 year old again in the point, Cyril minus six. So the other bona relying is this one. So now we have the three boundary lights. And now that we have all the boundary lines, we need to draw the region that we present each of the inequalities. And to do that, we're gonna use a test point. The test point should not belong. Toe any off the regions so we can use Let's say the point with coordinates 11 So we go with the first inequality X minus one less, earning more than six. And if we replace our test point into this inequality, we get one minus one less unequal than six and Since this is cereal, the inequality is true. So the test point belong to the region given by the first inequality. So this is going to be the first region correspondent to the red boundary, lying now for the second inequality X plus to why, lesson or equal than six. If we replace our test point, we will get one plus two listening within six. And since is this tree? This is true, and that means that our test point belongs toe the region given by the blue line or the second inequality. So must be something like this. And the third inequality, expletive and cereal. Good people don't see well, uh, we have that if we replace one greater and it wouldn't zero, and that's true. So that means that our test point also belong to the region. Give it by the third inequality or the green bound every night. Now the solution region is intersection off the three regions, so it's going to be that triangle there. And since each of the three boundary lines are solid, their solution regions also contains two pounder. Relax

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